Final answer:
To determine the number of roots on the imaginary axis for the given polynomial, more complex methods like factoring or numerical analysis would be needed, which are not provided in the original question. The Routh-Hurwitz criterion can find roots with positive real parts but not those on the imaginary axis.
Step-by-step explanation:
The student's question requires finding the number of roots on the imaginary axis of the s-plane for the polynomial function s⁶ + 2s⁵ + 8s⁴ + 12s³ + 20s² + 16s + 16. We can use the Routh-Hurwitz stability criterion to determine the number of roots with positive real parts. However, this method does not directly tell us the roots on the imaginary axis. To find roots on the imaginary axis specifically, we would need additional methods like factoring, synthetic division, or graphical approaches.
Since the question is likely about control systems or signal processing, a subject that often deals with such polynomials, we might also look for purely imaginary roots. If the polynomial has complex roots, they will occur in conjugate pairs, meaning that for any root at Bi, where B is a real number and i is the square root of -1, there will be a conjugate root at -Bi. Purely imaginary roots will be numbers where the real part is zero. Such roots would be considered critically stable in control system terminology, lying exactly on the imaginary axis and thus not contributing to the system's instability nor to its stability.